1. Make the kimchi: Cut cabbage in half lengthwise, then cut crossways into 1-inch pieces. In a large bowl, toss with onions, carrots, salt, and 1 tablespoon sugar. Refrigerate for 30 minutes.
2. Meanwhile, in a blender, combine garlic, ginger, gochujang, fish sauce, soy sauce, shrimp, and remaining sugar. Process into a paste, then add water, 1 tablespoon at a time, until thinned. Aggressively massage mixture into vegetables, then pack into a glass container, cover, and let ferment until bubbles subside, about 5 days. Transfer to a clean container and refrigerate for 2 days before eating.
3. Make the sliders: Season patties with salt and pepper and cook in a skillet or on a grill to desired internal temperature. Lightly toast buns, then top with patty, kimchi, bacon, and mayo.